This limit does not exist. For instance, we can evaluate this limit along any linear path by substituting a general linear equation: y=mx where m∈R.

Now notice that \(\displaystyle{\frac{{{x}^{{{4}}}+{4}{y}{\left\lbrace^{2}\right\rbrace}}}{{{x}^{{{2}}}+{2}{y}^{{{2}}}}}}={\frac{{{x}^{{{2}}}+{4}{m}^{{{2}}}}}{{{1}+{2}{m}^{{{2}}}}}}\rightarrow{\frac{{{4}{m}^{{2}}}}{{{1}+{2}{m}^{{{2}}}}}}{a}{s}{x}\rightarrow{0}\)

What you notice from this is that you get a different limit value for each value of mm that you choose. In order for the limit to exist, we must have the same limiting value, no matter which path we take the limit on. Since this function give different limits for each linear equation, we can conclude that it doesn’t exist.

Now notice that \(\displaystyle{\frac{{{x}^{{{4}}}+{4}{y}{\left\lbrace^{2}\right\rbrace}}}{{{x}^{{{2}}}+{2}{y}^{{{2}}}}}}={\frac{{{x}^{{{2}}}+{4}{m}^{{{2}}}}}{{{1}+{2}{m}^{{{2}}}}}}\rightarrow{\frac{{{4}{m}^{{2}}}}{{{1}+{2}{m}^{{{2}}}}}}{a}{s}{x}\rightarrow{0}\)

What you notice from this is that you get a different limit value for each value of mm that you choose. In order for the limit to exist, we must have the same limiting value, no matter which path we take the limit on. Since this function give different limits for each linear equation, we can conclude that it doesn’t exist.